The original Seven Essays, in a cool, new outfit ...
Email - [email protected]

All posts in General Questions

Probability: Multiple choice questions

Categories: General Questions
Comments Off on Probability: Multiple choice questions

For questions 1-5 use the random variable X with values x = 2, 3, 4, 5 or 6 with P(x) = 0.05x.

1. Determine P (x = 4).
a. 0.05 b. 0.10 c. 0.15 d. 0.20

2. Find P (x >= 4).
a. 0.60 b. 0.45 c. 0.75 d. 0.55

3. What is P (2 < x <= 5)?
a. 0.70 b. 0.60 c. 0.45 d. 0.35

4. Give E (x).
a. 4.5 b. 4.75 c. 4.25 d. 5

5. Calculate sigma*x (note: see attachment).
a. 1.75 b. 2.121 c. 1.323 d. 3.063

For questions 6-10 use a binomial random variable X with n = 5 and p = 0.4.

6. Calculate the probability that x equals three.
a. 0.40 b. 0.913 c. 0.064 d. 0.2304
7. Determine the probability that x is at most two.
a. 0.337 b. 0.6826 c. 0.913 d. 0.317

8. Find the probability that x is at least two.
a. 0.0778 b. 0.663 c. 0.3370 d. 0.6826

9. What is the expected value of x?
a. 2 b. 2.5 c. 1.75 d. 2.25

10. Give the standard deviation of x.
a. 1.44 b. 1.09545 c. 1.2 d. 1

For questions 11-15 use a normal random variable X with mean sixty and standard deviation six.

11. Calculate the Z-score for x = 52.
a. 1.33 b. 0.4082 c. -1.33 d. 0.0918

12. Determine the value of x that is equivalent to a Z-score of 1.96.
a. 71.76 b. 61.96 c. 48.24 d. 76.71

13. Find the probability that x is between forty-five and seventy.
a. 0.9587 b. 0.0062 c. 0.0537 d. 0.9463

14. Give the probability that x is at most 78.5.
a. 0.999 b. 0.4990 c. 3.08 d. 0.001

15. What is the value of x such that P (X < x) is 0.3264?
a. -0.45 b. 57.3 c. 62.7 d. 0.45

1. Suppose that for a 5-year-old automobile, the probability the engine will need repair in year 6 is 0.3, while the probability that the tires need replacing in year 6 is 0.8. The probability that both the engine will need repair and the tires will need replacing in year 6 is 0.2. What is the probability that the tires will need to be replaced and the engine will need repair?

ANSWER AND EXPLANATION

2. Suppose that for a 5 year old automobile, the probability the engine will need repair in year 6 is 0.3, while the probability that the tires need replacing in year 6 is 0.8. The probability that both the engine will need repair and the tires will need replacing in year 6 is 0.2. If it is known that the tires will need replacing, what is the probability that the engine needs repair?

ANSWER AND EXPLANATION
17

Poisson distribution probabilities and recursion relationship

Categories: General Questions
Comments Off on Poisson distribution probabilities and recursion relationship

The Poisson distribution is given by the following
P(x,λ)=e ^ -λ * λ^x! x=0,1,2,3…..j…..
Where λ>0 is a parameter which is the average value μ in poisson distribution.

a) show that the maximum poisson probability P(x=j,λ) occurs at approximately the average value, that is λ=j if λ>1.
(hint: you can take the first order derivative of the natural log of poisson probability, P(x=j, λ) with respect to λ and set it equal to 0

b) show that when λ<1 the poisson probability is a monotonically decreasing function of j, i.e, P(0, λ)>P(1, λ)>P(2, λ)….P(j, λ)…. And never has a maximum value
(hint: you can use the recursion relationship of the poisson distribution to prove this statement)

Tchebychev’s inequality

Categories: General Questions
Comments Off on Tchebychev’s inequality

The Tchebychev inequality can also be stated in the following way:
For any random variable x with mean equal to μ and variance equal to Δ². The minimum probability of X belong to the interval X?[ μ-k, μ+k] is at least:
P( | X- μ|<k ≥ 1-( Δ/k²)

Suppose that the random variables x1, x2, x3… xn form a random sample of size n drawn from some unknown distribution, then the sample mean is expressed as:
<Xn>=(x1+x2+x3+….+xn)/n
The mathematical expectation of sample mean is equal to:
E[<Xn>]= μ
The variance of the sample mean is equal to:
Var[<Xn>=Δ²/n

Now we are applying the Tchebyshev inequality to the sample mean <Xn> to estimate the probabilities:
a) show that P(|<Xn>-μ|>k ≤ Δ²/(nk²)
b) show that when the sample size increases, the probability of <Xn> outside k units from the mean μ decreases and asymptotically approaches to 0
c) suppose we know the variance Δ²=4 and we don’t know μ and we have observed the data x1, x2, x3… xn. How large the sample size n is required in order to make sure the probability of estimated μ will satisfy the following condition:
P(|<Xn>-μ|>1) ≤ 0.01

Operations on sets: Example

Categories: General Questions
Comments Off on Operations on sets: Example

1) Let G be the event that a girl is born. Let F be the event that a baby over 5 pounds is born. Characterized the union and the intersection of the two events?

2) Consider the event that a player scores a point in a game against team A and the event that the same player scores a point in a game against team B. What is the union of the two events? What is the intersection of the two events?

3) A brokerage firm deals in stocks and bonds. An Analyst for the firm is interested in assessing the probability that a person who inquires about the firm will eventually purchase stock (event S) or bonds (event B). Define the union and the intersection of these two events.

4) A committee is evaluating six equally qualified candidates for a job. Only three of the six will be invited for an interview; among the chosen three, the order of invitation is of importance because the first candidate will have the best chance of being accepted, the second will be made an offer only in the committee should reject both the first and the second. How many possible ordered choices of three out of six candidates are there?

Statistics M2: Six comprehensive problems

Categories: General Questions
Comments Off on Statistics M2: Six comprehensive problems

1. An auto supply store uses a fixed order size with safety stock system to control the inventory of a type of motor oil it sells. Demand for the oil averages 20 units per day (7300 units per year) and the standard deviation of the daily demand is 4.3 units. For this item, the cost of placing each order is $35. The store purchases the oil at a price of $10 per unit and the annual holding cost is 27% of the purchase price.
a) What is the economic order quantity (EOQ) for this item?
b) Calculate the total annual cost (purchase, ordering, and holding) if the store uses EOQ as its order size.
c) The lead time for ordering the oil is 14 days. The store wants the probability of not running out during lead time to be 0.90 (i.e., cycle service level=0.90). what should the reorder point be?
2. Toppers Food Market is an up-scale supermarket with its own in-store bakery. Every morning, Toppers must decide how many trays of its specialty bread to bake. Toppers earns a profit of $15.40 on each tray of bread they can sell on the day it is baked. Bread not sold on the day it is baked is discarded and Toppers incurs a loss of $6.60 per tray of bread they have to discard. Estimates of the daily demand of fresh bread are given in the table below.

How many trays of bread should toppers bake each morning?
Demand
(Trays) Probability of Selling
Exactly N Trays
5 0.20
6 0.20
7 0.25
8 0.15
9 0.15
10 0.05
11 0.00

3. A firm makes two end products. Product A is assembled from 3 units of component C and 1 unit of component E. Product B is assembled from 1 unit of component C and 2 units of component D. Each unit of component D is assembled from 2 units of E.
a) Draw the product structure trees from A and B.
b) Using the information below, complete the MRP schedules for all items. Note all of the Gross Requirements for end products A and B are shown in their schedules.

Item On Hand Lead Time Lot Sizing Rule
A 25 2 weeks FOQ = 50
B 0 1 week Lot for Lot
C 200 3 weeks FOQ = 200
D 0 1 week Lot for Lot
E 10 1 week FOQ = 50

Week
1 2 3 4 5 6 7
Gross Req’ts 30 10
A On Hand
Net Req’ts
P.O. Receipts
P.O. Releases

Gross Req’ts 20 10
On Hand
B Net Req’ts
P.O. Receipts
P.O. Releases

Gross Req’ts
On Hand
C Net Req’ts
P.O. Receipts
P.O. Releases

Gross Req’ts
On Hand
D Net Req’ts
P.O. Receipts
P.O. Releases

Gross Req’ts
On Hand
E Net Req’ts
P.O. Receipts
P.O. Releases

4. A printing company has five printing jobs to process through its largest printing press. The processing times and due dates for the five jobs are shown in the following table. The firm operates seven days per week.
Job Processing Time
(days) Due Date
(days hence)
A 13 15
B 8 20
C 2 5
D 15 25
E 5 30

The firm`s objective is to minimize the average tardiness for this set of jobs.
Note: the term tardiness is used here , rather than lateness, to indicate that there is no credit for a job completed early. A job completed before its due date simply has tardiness value of zero. In other words, lateness and tardiness are the same if the job is completed after the due date, but different if the job is completed before the due date.
a) Schedule the jobs using the earliest due date (EDD) priority rule and calculate the resulting mean tardiness for this set of jobs.
b) Schedule the jobs using the slack time remaining (STR) priority rule and calculate the resulting mean tardiness for this set of jobs.
c) Which of the two priority rules was best in this particular case?
5. A firm sells 30 kilogram bags of Portland cement to Home Depot and other retailers who sell them to small contractors and do it yourselves. Automatic machinery is used to fill the bags. Periodically, samples of size 8 are taken from the filling line and carefully weighed. While the process was in control, the average of the sample means was found to be 31.42 kilograms and the average of the sample ranges was found to be 0.81 kilograms
N A2 D4 D4 N A2 D3 D4
2 1.88 0 3.27 7 0.42 0.08 1.92
3 1.02 0 2.57 8 0.37 0.14 1.86
4 0.73 0 2.28 9 0.34 0.18 1.82
5 0.58 0 2.11 10 0.31 0.22 1.78
6 0.48 0 2.00 11 0.29 0.26 1.74

Calculate 3σ control limits for mean and range charts to monitor the filling process.

6. An insurance firm is concerned about the number of claims forms with missing or incorrect information. Analysis of a large number of forms has shown that 8.4% of the forms are defective in some way. To monitor the process, a sample of 300 forms will be taken each day and checked for errors.
a) Calculate 3 σ control limits for a P-chart to monitor the process.
b) The numbers of defective forms found in the five most recent samples are given in the following table. Complete the chart. Plot the results of the five samples.
Sample # 1 2 3 4 5
Number of Defective Forms 33 18 30 9 27

Sample #
1 2 3 4 5

c) Is the process in control?

7. Analyze and solve the following project network. The numbers in the top line of the boxes represent the time in days required to perform each activity.
a) Using the two-step procedure consisting of forward and backward passes; determine how long it will take to complete the entire project.

3

2

2

0
0 0

0

Start Finish
Duration
ES EF
LS LF

7

3

5

b) Calculate the slack times. Which activities are on the critical path?
c) If activity C takes 3 days to complete and activity E takes 3 days to complete, will the completion of the project be delayed?

8. The following diagram shows the project network for the development of a radically new product. There is considerable uncertainty in the length of time required to complete each of the activities on the network, so probabilistic activity times have been used. The expected activity time (ET) and the variance of the activity time (VAR) for each activity are given in the table.

Activity ET
(Weeks) VAR
(Weeks)
A 14 3.67
B 9 1.33
C 13 4.33
D 15 2.00

Start Finish
a) What is the expected completion time for the project?
b) Estimate the probability that the project will be completed in 31 weeks.

Probability

Categories: General Questions
Comments Off on Probability

If P is a normally distributed random variable with a mean of 50 and a standard deviation of 2, what is the probability that P is between 47 and 54?

Problems using Probability Distribution of Random Variables

Categories: General Questions
Comments Off on Problems using Probability Distribution of Random Variables

A real estate agent has four houses to sell before the end of the month by contacting prospective customers one by one. Each costumer has an independent 0.24 probability of buying a house on being contacted by the agent.

a) If the agent has enough time to contact only 15 customers, how confident can she be of selling all four houses within the available time?
b) If the agent wants to be a at least 70% confident of selling all the houses within the available time , at least how many customer should she contact?(if necessary, extend the template downward to more rows)
c) What minimum value of p will yield 70% confidence of selling all four houses by contacting at most 15 costumers?

3-52
Laptop computers produced by a company have an average life of 38.36 months. Assume that the life of a computer is exponentially distributed (which is a good assumption)
a) What is the probability that a computer will fail within 12 months?
b) If the company gives a warranty period of 12 months, what proportion of computers will fail during the warranty period?
c) Based on the answer (b), would you say the company can afford to give a warranty period of 12 months?
d) If the company wants not more than 5% of the computers to fail during the warranty period, what should be the warranty period?
e) If the company wants to give a warranty period of three months and stills wants not more than 5% of the computers to fail during the warranty period, what should be the minimum average life of the computers?

3-69
The number of orders for installation of a computer information system arriving at an agency per week is a random variable X with the following probability.
X P(x)
0 0.10
1 0.20
2 0.30
3 0.15
4 0.15
5 0.05
6 0.05
a) Prove that P (X) is a probability distribution
b) Find the cumulative distribution function of X
c) Use the cumulative distribution function to find probabilities P (2 < X ≤ 5) , P (3 ≤ X ≤ 6) and P ( X> 4).
d) What is the probability that either four or five orders will arrive in a given week?
e) Assuming independence of weekly orders, what is the probability that three orders will arrive next week and the same number of orders the following week?
f) Find the mean and the standard deviation of the number of weekly orders.

3-77
Suppose that 5 of a total of 20 company accounts are in error. An auditor selects a random sample of 5 out of the 20 accounts. Let X be the number of accounts in the sample that are in error. Is X binomial? If not, what distribution does it have? If not, what distribution does it have? Explain.